SPHERICAL ABERRATION FOR A SYMMETRICAL OPTICAL SYSTEM. 
65 
this, however, is contributed by three cases, namely : M = 3 and 2 for lens I where 
the aberrations are very large and differ very widely from the usual first and second 
order approximations, so that, although the error of the formula approaches 2 per cent., 
it nevertheless represents a great improvement upon these approximations; and 
M = 0'5 for lens 5, which corresponds to extreme curvature and highest inclination, 
so that one of the angles of refraction is as great as 48|- degrees. Even here the 
table below shows that the formula is an appreciable improvement on the usual 
second-order approximation. If these three cases are omitted, the mean percentage 
error works out to be about 0'21, so that in general the formula determines the 
longitudinal aberration correct to about 1 part in 500. 
It is interesting to note what the usual first and second order approximations lead 
to in a few cases. 
First order 
approximation. 
Percentage 
error. 
Second order 
approximation. 
Percentage 
error. 
Lens 1, M = 3 . . . 
- 1-34012 
83-0 
-2-45432 
68-8 
„ 1, M = 2 ... 
-0-48545 
68-1 
-0-82104 
46-1 
„ 1, M = -2 . . . 
-1-348142 
34-1 
-0-88456 
12-0 
„ 2, M = 3 ... 
-0-67356 
40-3 
-0-94436 
16-3 
„ 5, M = 0-5. . . 
-0-12136 
19-5 
-0-14307 
5-1 
This gives a measure of the numerical improvement effected by the fractional formula 
whenever the usual method of approximation is seriously out, even though in none of 
the cases above does the convergency of the series actually fail. 
In the above the series are in powers of tan /3 2 . Had they been taken in powers of 
tan a 2 , as is frequently done, the first and second order approximations would have 
been far worse. 
One interesting outcome of these calculations relates to the relative importance of 
the terms in E£ 4 4 and A tf The ratio E t^/A is small in every case taken (of course 
these exclude the neighbourhood of points where A = 0, where naturally E becomes 
of great importance). But for the set of magnifications taken, the greatest ratio of 
the second term to the first is less than 0‘03 and the mean value of this ratio is only 
0'0082, so that, in fact, the E term—although so complicated algebraically—does not 
exercise any great influence numerically. 
This is important, as it shows that, at any rate for lenses, it does not require to be 
computed with anything like the same order of accuracy which is needed for A and B. 
§ 14. The Singular Inclination and Convergency Factor for any System. 
Referring again to fig. 2 we see that X = a 2 and F o I 0 = —Ax 2 for rays proceeding 
through the system reversed and initially parallel. 
Thus using accents, as before, to denote the coefficients and inclinations for 
